# CHAPTER 8General Linear Transformations

## INTRODUCTION

In earlier sections we studied linear transformations from ${R}^{n}$ to ${R}^{m}$. In this chapter we will define and study linear transformations from a general vector space $V$ to a general vector space $W$. The results we will obtain here have important applications in physics, engineering, and various branches of mathematics.

## 8.1 General Linear Transformations

Up to now our study of linear transformations has focused on transformations from ${R}^{n}$ to ${R}^{m}$. In this section we will turn our attention to linear transformations involving general vector spaces. We will illustrate ways in which such transformations arise, and we will establish a fundamental relationship between general $n$-dimensional vector spaces and ${R}^{n}$.

### Definitions and Terminology

In Section 1.8 we defined a matrix transformation ${T}_{A}:{R}^{n}\to {R}^{m}$ to be a mapping of the form

${T}_{A}\left(\mathbf{x}\right)=A\mathbf{x}$

in which $A$ is an $m×n$ matrix. We subsequently established in Theorem 1.8.3 that the matrix transformations are precisely the linear transformations from ${R}^{n}$ to ${R}^{m}$, that is, the transformations with the linearity properties

$\begin{array}{ccc}\hfill T\left(\mathbf{u}+\mathbf{v}\right)=T\left(\mathbf{u}\right)+T\left(\mathbf{v}\right)\hfill & \hfill \text{and}\hfill & \hfill T\left(k\mathbf{u}\right)=kT\left(\mathbf{u}\right)\hfill \end{array}$

We will use these two properties as the starting point for defining more general linear transformations. ...

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